What is the derivative of tan^7(x^2)?

1 Answer
Aug 14, 2017

d/(dx) [tan^7(x^2)] = color(blue)(14xtan^6(x^2)sec^2(x^2)

Explanation:

We're asked to find the derivative

d/(dx) [tan^7(x^2)]

We can first use the chain rule:

d/(dx) [tan^7(x^2)] = d/(du) [u^7] (du)/(dx)

where

  • u = tan(x^2)

  • d/(du) [u^7] = 7u^6:

= 7d/(dx)[tan(x^2)]tan^6(x^2)

Using the chain rule again:

d/(dx) [tan(x^2)] = d/(du) [tanu] (du)/(dx)

where

  • u = x^2

  • d/(du) [tanu] = sec^2u:

= 7d/(dx)[x^2]sec^2(x^2)tan^6(x^2)

Use the power rule on the x^2 term:

d/(dx) [x^n] = nx^(n-1)

where

  • n = 2:

= 7(2x)sec^2(x^2)tan^6(x^2)

color(blue)(ulbar(|stackrel(" ")(" "= 14xsec^2(x^2)tan^6(x^2)" ")|)